1. Enhanced conformational sampling via very large time-step molecular dynamics, novel variable transformations and adiabatic dynamics Mark E. Tuckerman Dept. of Chemistry and Courant Institute of Mathematical Sciences New York University, 100 Washington Sq. East New York, NY 10003

2. Acknowledgments Zhongwei Zhu Peter Minary Lula Rosso Jerry Abrams NSF - CAREER NYU Whitehead Award NSF – Chemistry, ITR Camille and Henry Dreyfus Foundation Students past and present Postdocs Dawn Yarne Radu Iftimie Collaborators Glenn Martyna Christopher Mundy Funding

3. Talk Outline Very large time-step multiple time scale integration that avoids resonance phenomena. Novel variable transformations in the partition function for enhancing conformational sampling. Adiabatic decoupling along directions with high barriers for direct computation of free energies.

4. Multiple time scale (r-RESPA) integration MET, G. J. Martyna and B. J. Berne, J. Chem. Phys. 97, 1990 (1992)

5. Resonance Phenomena Large time step still limited by frequency of the fast force due to numerical artifacts called resonances. Problematic whenever there is high frequency weakly coupled to low frequency motion Biological Force Fields Path integrals Car-Parrinello molecular dynamics

6. Illustration of resonance A. Sandu and T. S. Schlick, J. Comput. Phys. 151, 74 (1999)

7. Illustration of resonance (cont’d) Depending on Δt, eigenvalues of A are either complex conjugate pairs Note: det(A) = 1 or eigenvalues are both real Leads to resonances (|Tr(A)| → 2) at Δt = nπ/ω

8. Resonant free multiple time-scale MD Resonance means time steps are limited to 5-10 fs for most problems. Assign time steps to each force component based on intrinsic time scale. Prevent any mode from becoming resonant via a kinetic energy constraint. Ensure ergodicity through Nosé-Hoover chain thermostatting techniques. P. Minary, G. J. Martyna and MET, Phys. Rev. Lett. 93, 150201 (2004).

9. Review of isokinetic dynamics Constraint the kinetic energy of a system: Introduce constraint via a Lagrange multiplier: Derivative of constraint yields multiplier: Partition function generated:

10. Review of Nosé-Hoover Equations For each degree of freeom with coordinate q and velocity v,

11. New equations of motion (Iso-NHC-RESPA) Couple each degree of freedom to the first element of L NHCs of length M Ensures the constraint:is satisfied.

12. Classical non-Hamiltonian statistical mechanics General equations of motion: Consider a solution: If the equations are non-Hamiltonian. Κ(x) called the compressibility of the equations. In order to generalize Liouville’s theorem, we need to determine: Tuckerman, Mundy, Martyna, Europhys. Lett. 45, 149 (1999); Tuckerman, et al. JCP 115, 1678 (2001).

13. Classical non-Hamiltonian statistical mechanics

14. Solution: Note that for Hamiltonian systems, κ(x)=0 and J(x t,x 0 )=1. Define: Then: Whence:

15. Classical non-Hamiltonian statistical mechanics Define a metric factor: In addition, suppose the dynamical equations have N c conservation laws of the form: Then, the dynamical system, assuming ergodicity, will generate a “microcanonical” ensemble whose partition function is: Also, assume equilibrium conditions, i.e. that and the phase space distribution has no explicit time dependence.

16. Phase space distribution For the Iso-NHC-RESPA method: Metric Factor: For the present system:

17. Integration of the equations Liouville operator decomposition: Factorized propagator:

18. Numerical illustration of resonance Harmonic oscillator with quartic perturbation

19. Flexible TIP3P water Intramolecular forces Short-range forces cutoff = 5Å Long-range forces 10 Å + Ewald

20. HIV-1 Protease in vacuo 1.52.53.54.5 r CH (A) g(r) 0.9 1.01.11.2

22. Conformational sampling in Biophysics “Ab initio” protein/nucleic acid structure prediction: Sequence → Folded/active structure. Enzyme catalysis. Drug docking/Binding free energy. Tracking motion water, protons, other ions.

23. Unfolded State Native State Misfolded State

24. The conformational sampling problem Find low free energy structures of complex molecules Sampling conformations described by a potential function: V(r 1,…,r N ) Protein with 100 residues has ~10 50 conformations. “Rough free energy landscape” in Cartesian space. Solution: Find a smoother space in which to work. Z. Zhu, et al. Phys. Rev. Lett. 88, art. No. 100201 (2002) P. Minary, et al. (in preparation)

28. REPSWA (Reference Potential Spatial Warping Algorithm)

29. No TransformationTransformation

30. Barrier Crossing Transformations (cont’d) ‘ ‘ ‘ ‘

31. V ref ( Φ )

35. A 400-mer alkane chain

36. RIS Model value: 10 No TransformationTransformation

40. Model sheet protein No Transformation Parallel Tempering Dynamic transformation

41. No TransformationsTransformations

43. L. Rosso, P. Minary, Z. Zhu and MET, J. Chem. Phys. 116, 4389 (2000)

45. Conformational sampling of the solvated alanine dipeptide ψφ AFED T φ,ψ = 5T, M φ,ψ = 50M C 4.7 ns Umbrella Sampling 50 ns CHARM22 αRαR β [L Rosso, J. B. Abrams and MET (in preparation)]

46. Conformational sampling of the gas-phase alanine dipeptide ψφ AFED T φ,ψ = 5T, M φ,ψ = 50M C 3.5 ns Umbrella Sampling 35 ns CHARM22 β

47. Conformational sampling of the gas-phase alanine tripeptide AFED T φ,ψ = 5T, M φ,ψ = 50M C 4.7 ns Umbrella Sampling 50 ns β C ax 7 φ1φ1 ψ1ψ1 ψ2ψ2 φ2φ2

48. Conformational sampling of the solvated alanine tripeptide

51. Closed ~ 5Å Open ~15Å R Protonation state of the active site important in drug binding

52. RIS Model value: 14 No Transformation Transformation

54. Water Number Density (Å ) Protease alone: 0.024 Protease + drug: 0.015 Protease alone Protease + drug Z. Zhu, D. I. Schuster and MET, Biochemistry 42, 1326 (2003) Avg. cavity dimensions (Å) HeightWidth PR alone 20.7 12.3 PR + drug 19.2 17.3 PR + Saq. 20.2 15.1 Bulk water: 0.033 -3

55. Conclusions Isokinetic-NHC-RESPA method allows time steps as large as 100 fs to be used in typical biophysical problems. Variable transformations lead to efficient MD scheme and exactly preserve partition function. Speedups of over 10 6 possible in systems with many backbone dihedral angles. Trapped states are largely avoided. Future: Combine variable transformations with Iso-NHC-RESPA Future: Develop variable transformations for ab initio molecular dynamics, where potential surface is unknown.